Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 71, pp. 1-32.

Title: Integrable nonlinear perturbed hierarchies of NLS-mKdV equation and soliton solutions
       
Authors: Qiulan Zhao    (Shandong Univ. of Science and Tech., Qingdao, Shandong, China)
         Hongbiao Cheng (Shandong Univ. of Science and Tech., Qingdao, Shandong, China)
         Xinyue Li      (Shandong Univ. of Science and Tech., Qingdao, Shandong, China)
         Chuanzhong Li  (Shandong Univ. of Science and Tech., Qingdao, Shandong, China) 

Abstract:
We propose three spectral problems for NLS-mKdV equation by combining three
integrable coupling ways. Then we obtain three nonlinear perturbation terms
to derive three integrable nonlinear perturbed hierarchies of the NLS-mKdV equation.
We proved the Lax integrability of the integrable nonlinear perturbed hierarchies.
On the basis of a special orthogonal group, we prove the Liouville integrability of
a third-order integrable nonlinear  perturbed hierarchy of NLS-mKdV equation by
deriving its bi-Hamiltonian structures.
We build three Darboux matrices for constructing the Darboux transformations
of the first two equations. As applications of the Darboux transformation,
we present explicit solutions of these equations, three-dimensional plots, 
and density
profiles the evolution of solitary waves.

Submitted Submitted April 29, 2022. Published October 13, 2022.
Math Subject Classifications: 35Q51, 37K10.
Key Words: Integrable perturbed hierarchies; nonlinear perturbation terms;
    Darboux transformation; soliton solutions.